III Small-Scale Intermittency & Anomalous Scaling We have seen that turbulent energy dissipation non-vanishing as Re !1 requires that ⇣ p/3 for p ≥ 3. The K41 theory assumes the “minimal singularity” sufficient to dissipate energy, or ⇣ = p/3 for all p. However, other possibilities are allowed by the above estimate! In this set of notes we consider the subject of turbulent scaling laws and their relation to turbulent energy cascade. (A) A Simple Model of Energy Dissipation: Burgers Equation In this section we consider a simple 1-dimensional PDE model that has non-vanishing energy dissipation for Re !1 but for which K41 theory fails . It is a useful counterexample! The model is the 1-dimensional Burgers equation for a velocity field u(x, t): @ t u + u@ x u = ⌫@ 2 x u. It can also be written as @ t u + @ x ( 1 2 u 2 )= ⌫@ 2 x u so that it corresponds to a conservation of “momentum” R u(x, t)dx. As a simple prototype of turbulence, it was first proposed by J. M. Burgers, “ A mathematical model illustrating the theory of turbulence,” Adv. Appl. Mech. 1, 171-199 (1948). The “energy” E(t)= 1 2 R u 2 (x, t)dx is also conserved in the limit ⌫ ! 0, i.e. for “ideal” Burgers equation, since @ t ( 1 2 u 2 )+ @ x [ 1 3 u 3 - ⌫@ x ( 1 2 u 2 )] = -⌫ (@ x u) 2 . Formally, ⌫ (@ x u) 2 ! 0 as ⌫ ! 0. However, this is NOT what occurs! Consider a simple exact solution of 1-D burgers: u(x, t)= 1 t [x - L tanh( Lx 2⌫t )]. This seems to have been first written down in 1
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III Small-Scale Intermittency & Anomalous Scaling
We have seen that turbulent energy dissipation non-vanishing as Re ! 1 requires that
⇣ p/3 for p � 3.
The K41 theory assumes the “minimal singularity” su�cient to dissipate energy, or ⇣ = p/3
for all p. However, other possibilities are allowed by the above estimate! In this set of notes we
consider the subject of turbulent scaling laws and their relation to turbulent energy cascade.
(A) A Simple Model of Energy Dissipation: Burgers Equation
In this section we consider a simple 1-dimensional PDE model that has non-vanishing energy
dissipation for Re ! 1 but for which K41 theory fails. It is a useful counterexample! The
model is the 1-dimensional Burgers equation for a velocity field u(x, t):
@tu+ u@xu = ⌫@2xu.
It can also be written as
@tu+ @x(12u
2) = ⌫@2xu
so that it corresponds to a conservation of “momentum”Ru(x, t)dx. As a simple prototype of
turbulence, it was first proposed by J. M. Burgers, “ A mathematical model illustrating the
theory of turbulence,” Adv. Appl. Mech. 1, 171-199 (1948).
The “energy”
E(t) = 12
Ru2(x, t)dx
is also conserved in the limit ⌫ ! 0, i.e. for “ideal” Burgers equation, since
@t(12u
2) + @x[13u
3 � ⌫@x(12u
2)] = �⌫(@xu)2.
Formally, ⌫(@xu)2 ! 0 as ⌫ ! 0. However, this is NOT what occurs!
Consider a simple exact solution of 1-D burgers:
u(x, t) = 1t [x� L tanh( Lx2⌫t)].
This seems to have been first written down in
1
S. I. Soluyan & R. V. Khokhlov, Vestnik. Moscow State. Univ. Phys. Astron.
3 52-61(1961)
and is sometimes called the “Khokhlov saw-tooth solution”. The reason for the term “sawtooth”
is that
u⌫(x, t) ⇠ 1
t(x+ L) as x ! �1
u⌫(x, t) ⇠ 1
t(x� L) as x ! +1
with a discontinuity 4u = (2L)/t across the origin.
Plots of the Khokhlov sawtooth solution at fixed time for various viscosities
In the limit ⌫ ! 0, this becomes
u(x, t) =
8><
>:
(x+ L)/t �L x < 0
(x� L)/t 0 < x L
2
a function on the interval [�L,L] with u(±L, t) = 0 a sharp discontinuity of size 4u = (2L)/t
at x = 0. Such a discontinuity is called a shock or, in this case, a stationary shock, since it is
located at the same point x = 0 for all time t.
Now it is easy to see that there is nonvanishing mean energy dissipation in the limit that ⌫ ! 0.
For example, at ⌫ = 0, using the above explicit formula, it is easy to see that
1
2hu2(t)i =
1
2L
Z L
�L
1
2u2(x, t)dx
=1
2L
Z L
0
✓x� L
t
◆2
dx =1
6
✓L
t
◆2
so that
h"(t)i = � ddt
12hu2(t)i = 1
3L2
t3= (4u)2
12t > 0!!!
Alternatively, one can consider the viscous dissipation
"⌫(x, t) = ⌫|@xu⌫(x, t)|2
using
@xu⌫(x, t) =1t � L2
2⌫t2 sech2( Lx2⌫t)
so that ⌫ ⌧ L2/t, with L and t fixed,
"⌫(x, t) ⇡ L4
4⌫t4 sech4( Lx2⌫t)
The energy dissipation becomes very large ⇠ L4
⌫t4in a small region of size ⇠ ⌫t/L. Using the
simple integralR +1�1 sech4u du = 4
3 , we again finds that
h"⌫(t)i =1
2L
Z L
�L"⌫(x, t)dx
⇠= 1
2L· L4
4⌫t4· 2⌫tL
· 43
for ⌫ ! 0
=1
3
L2
t3or
(4u)2
12t
Again, the limit as ⌫ ! 0 is positive! This is exactly like the experiments & simulations for
real fluids!
3
Plots of energy dissipation in the Khokhlov sawtooth solution for various viscosities
All of our previous theory applies to this problem. E.g. we may consider the coarse-grained
equation
@tu` + @x(12 u
2` + ⌧`) = ⌫@2
xu`
with
⌧` =1
2[(u2)` � u2` ]
=1
2[h(�u)2i` � h�ui2` ] = O(�u2(`))
The large-scale energy balance is
@t(12 u
2` ) + @x[
13 u
3` + ⌧`u` � ⌫@x(
12 u
2` )] = ⌧`(@xu`)� ⌫(@xu`)2.
For fixed ` we see again that ⌫(@xu`)2 = O(⌫�2u(`)`2
), which goes to zero as ⌫ ! 0. Large-scale
“dissipation” must come from
4
⌧`@xu` = O( �u3(`)` )
from which we can deduce, as before, that
⇣p p/3 for p � 3
However, K41 theory does not work here! For the limiting shock profile at ⌫ ! 0:
u(x, t) =
8><
>:
(x+ L)/t �L x < 0
(x� L)/t 0 < x L,
periodically extended to R with period 2L, we can see that (with ` > 0)
u(x+ `, t)� u(x, t) =
8><
>:
`/t if 0 /2 [x, x+ `]
(2L+ `)/t if 0 2 [x, x+ `]=)
h|�u(`)|pi =1
2L
Z L
�L|u(x+ `)� u(x)|pdx
= (1� `
2L) · (`
t)p +
`
2L· (2L+ `
t)p
⇠ (4u)p
8><
>:
( `2L)
p 0 < p < 1
`2L p � 1
for ` ⌧ L
Thus,
⇣p =
8><
>:
p 0 < p < 1
1 p � 1
Of course, 1 p/3 for p � 3, so that our inequality is verified that ⇣p p/3 but only for p = 3
is ⇣p = p/3!
The problem is that K41 theory assumes that h = 13 at every point of space and that is not
what happens here. Instead, there is one point (x = 0) where
�u(`, x) ⇠ 4u ⇠ `0 for all ` < L
and, at every other point,
�u(`, x) ⇠ `/t ⇠ `1
for su�ciently small `. This is an extreme example of small-scale intermittency, in which
velocity increments are “spotty” in space, big in some places and small in others.
5
Incidentally, it is known that the above features that we have seen in a simple specific solution
of Burgers equation are, in fact, generic for that dynamics. Except for fields in which @xu > 0
everywhere, the Burgers solutions always develop shocks that become exact discontinuities in
the limit ⌫ ! 0 and these dissipate a finite amount of energy that does not vanish as ⌫ ! 0.
The scaling exponents ⇣p that we determined above are also universal to a wide class of initial
data and forcing schemes.
For more information about Burgers equation and “Burgulence,” see :
J. M. Burgers, The Non-Linear Di↵usion Equation: Asymptotic Solutions &
Statistical Problems, (Springer, Boston, 1974)
W. E. K. Khanin, A. Mazel and Y. Sinai, “Invariant Measures for Burgers Equa-
tion with Stochastic Forcing,” Ann. Math. 151 (3): 817-960 (2000)
and a recent review article
J. Bec & K. Khanin, “Burgers Turbulence,” Physics Reports 447 1-66 (2007)
Before we leave the topic of Burgers equation, there is one last important remark we wish to
make. Consider again the limiting shock profile
u(x, t) =
8><
>:
(x+ L)/t �L x < 0
(x� L)/t 0 < x Lperiodic on [�L,L]
These results are just special cases of a general theory for Burgers equation (and for a whole
class of scalar conservation laws in 1-dimension). It is known that the solution u⌫(x, t) of
@tu⌫(x, t) + u⌫(x, t)@xu⌫(x, t) = ⌫@2xu
⌫(x, t)
converges in the sense of distributions u⌫(x, t) ! u(x, t) as ⌫ ! 0 to a solution of the inviscid
equation
@tu(x, t) + @x(12u
2(x, t)) = 0
which satisfies
@t(12u
2(x, t)) + @x(13u
3(x, t)) 0,
6
both in the sense of distributions. It is quite easy to check that this profile satisfies the inviscid
Burgers equation
@tu(x, t) + u(x, t)@xu(x, t) = 0
at every spacetime point (x, t) with x 6= 0. The problem at x = 0 is that a derivative (@xu)
does not even exist in the classical sense. However, it is not hard to check that the limiting
shock profile satisfies everywhere in space and for all ` > 0 the coarse-grained equations.
@tu` + @x(12u
2)`= 0 (*)
or, with a mere change of notation,
@tu` + @x(12 u
2` + ⌧`) = 0
with ⌧` = 12 [(u
2)` � u2` ], where all derivatives of the coarse-grained variables do exist in a
classical sense. Here the initial data u`(x, 0) for (*) are also coarse-grained. An equivalent,
more conventional formulation of (*) is obtained by noting that derivatives can be taken to
exist in the sense of distributions and it can easily be verified thatR10 dt
R L�L dx[u(x, t)@t'(x, t) +
12u
2(x, t)@x'(x, t)] +R L�L dx u(x, 0)'(x, 0) = 0
for all C1 functions ' on [�L,L]⇥ [0,1] with compact support (in time). Thus, u(x, t) satifies
the inviscid Burgers equation in the sense of distributions, with initial data u(x, 0). However,
it is also easy to check that energy is not conserved and, in fact, that
@t(12u
2(x, t)) + @x(13u
3(x, t)) = � (�u)3
12 �(x) 0 (**)
in the sense of distributions.
Furthermore, for a given initial datum u(x, 0) this distributional solution is unique. Note
that, in general, such “weak” or distributional solutions are not unique and there can be more
than one solution even for the same initial data. See examples in
P. D. Lax, “Hyperbolic systems of conservation laws, II.” Commun. Pure Appl.
Math. 10 537-566 (1957)
It does not make sense to talk about the solution of the inviscid equation — it is an ill-defined
concept! It is only weak solutions with the energy inequality (**) that are unique. It is known
7
that any hyperviscous regularized solution
@tu✏(x, t) + @x(12 |u✏(x, t)|2) = �✏(�@2
x)pu✏(x, t), ✏ > 0
with p � 1 converges as ✏ ! 0 to the same u(x, t) as for p = 1. See:
E. Tadmor, “Burgers equation with vanishing hyperviscosity,” Comm. Math.Sci.
2(2), 317-324(2004).
However, a dispersive regularization such as the famous Korteweg-de Vries (KDV) equation
@tu✏(x, t) + @x(12 |u✏(x, t)|2) = ✏@3
xu✏(x, t), ✏ > 0
which describes weakly nonlinear shallow water waves, has a completely di↵erent class of solu-
tions, even for the same initial data u(x, 0). See
P. D. Lax and C. D. Levermore, “The small dispersion limit for the KDV equa-